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Theorem tfinds 4806
Description: Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. Theorem Schema 4 of [Suppes] p. 197. (Contributed by NM, 16-Apr-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypotheses
Ref Expression
tfinds.1  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
tfinds.2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
tfinds.3  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
tfinds.4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
tfinds.5  |-  ps
tfinds.6  |-  ( y  e.  On  ->  ( ch  ->  th ) )
tfinds.7  |-  ( Lim  x  ->  ( A. y  e.  x  ch  ->  ph ) )
Assertion
Ref Expression
tfinds  |-  ( A  e.  On  ->  ta )
Distinct variable groups:    x, y    x, A    ch, x    ta, x    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)    ch( y)    th( x, y)    ta( y)    A( y)

Proof of Theorem tfinds
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 tfinds.2 . 2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
2 tfinds.4 . 2  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
3 dflim3 4794 . . . . 5  |-  ( Lim  x  <->  ( Ord  x  /\  -.  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
43notbii 288 . . . 4  |-  ( -. 
Lim  x  <->  -.  ( Ord  x  /\  -.  (
x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
5 iman 414 . . . . 5  |-  ( ( Ord  x  ->  (
x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) )  <->  -.  ( Ord  x  /\  -.  ( x  =  (/)  \/ 
E. y  e.  On  x  =  suc  y ) ) )
6 eloni 4559 . . . . . . 7  |-  ( x  e.  On  ->  Ord  x )
7 pm2.27 37 . . . . . . 7  |-  ( Ord  x  ->  ( ( Ord  x  ->  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) )  -> 
( x  =  (/)  \/ 
E. y  e.  On  x  =  suc  y ) ) )
86, 7syl 16 . . . . . 6  |-  ( x  e.  On  ->  (
( Ord  x  ->  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) )  ->  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) ) )
9 tfinds.5 . . . . . . . . 9  |-  ps
10 tfinds.1 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
119, 10mpbiri 225 . . . . . . . 8  |-  ( x  =  (/)  ->  ph )
1211a1d 23 . . . . . . 7  |-  ( x  =  (/)  ->  ( A. y  e.  x  ch  ->  ph ) )
13 nfra1 2724 . . . . . . . . 9  |-  F/ y A. y  e.  x  ch
14 nfv 1626 . . . . . . . . 9  |-  F/ y
ph
1513, 14nfim 1828 . . . . . . . 8  |-  F/ y ( A. y  e.  x  ch  ->  ph )
16 vex 2927 . . . . . . . . . . . . 13  |-  y  e. 
_V
1716sucid 4628 . . . . . . . . . . . 12  |-  y  e. 
suc  y
181rspcv 3016 . . . . . . . . . . . 12  |-  ( y  e.  suc  y  -> 
( A. x  e. 
suc  y ph  ->  ch ) )
1917, 18ax-mp 8 . . . . . . . . . . 11  |-  ( A. x  e.  suc  y ph  ->  ch )
20 tfinds.6 . . . . . . . . . . 11  |-  ( y  e.  On  ->  ( ch  ->  th ) )
2119, 20syl5 30 . . . . . . . . . 10  |-  ( y  e.  On  ->  ( A. x  e.  suc  y ph  ->  th )
)
22 raleq 2872 . . . . . . . . . . . 12  |-  ( x  =  suc  y  -> 
( A. z  e.  x  [ z  /  x ] ph  <->  A. z  e.  suc  y [ z  /  x ] ph ) )
23 nfv 1626 . . . . . . . . . . . . . . 15  |-  F/ x ch
2423, 1sbie 2095 . . . . . . . . . . . . . 14  |-  ( [ y  /  x ] ph 
<->  ch )
25 sbequ 2117 . . . . . . . . . . . . . 14  |-  ( y  =  z  ->  ( [ y  /  x ] ph  <->  [ z  /  x ] ph ) )
2624, 25syl5bbr 251 . . . . . . . . . . . . 13  |-  ( y  =  z  ->  ( ch 
<->  [ z  /  x ] ph ) )
2726cbvralv 2900 . . . . . . . . . . . 12  |-  ( A. y  e.  x  ch  <->  A. z  e.  x  [
z  /  x ] ph )
28 cbvralsv 2911 . . . . . . . . . . . 12  |-  ( A. x  e.  suc  y ph  <->  A. z  e.  suc  y [ z  /  x ] ph )
2922, 27, 283bitr4g 280 . . . . . . . . . . 11  |-  ( x  =  suc  y  -> 
( A. y  e.  x  ch  <->  A. x  e.  suc  y ph )
)
3029imbi1d 309 . . . . . . . . . 10  |-  ( x  =  suc  y  -> 
( ( A. y  e.  x  ch  ->  th )  <->  ( A. x  e.  suc  y ph  ->  th ) ) )
3121, 30syl5ibrcom 214 . . . . . . . . 9  |-  ( y  e.  On  ->  (
x  =  suc  y  ->  ( A. y  e.  x  ch  ->  th )
) )
32 tfinds.3 . . . . . . . . . . 11  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
3332biimprd 215 . . . . . . . . . 10  |-  ( x  =  suc  y  -> 
( th  ->  ph )
)
3433a1i 11 . . . . . . . . 9  |-  ( y  e.  On  ->  (
x  =  suc  y  ->  ( th  ->  ph )
) )
3531, 34syldd 63 . . . . . . . 8  |-  ( y  e.  On  ->  (
x  =  suc  y  ->  ( A. y  e.  x  ch  ->  ph )
) )
3615, 35rexlimi 2791 . . . . . . 7  |-  ( E. y  e.  On  x  =  suc  y  ->  ( A. y  e.  x  ch  ->  ph ) )
3712, 36jaoi 369 . . . . . 6  |-  ( ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y )  -> 
( A. y  e.  x  ch  ->  ph )
)
388, 37syl6 31 . . . . 5  |-  ( x  e.  On  ->  (
( Ord  x  ->  ( x  =  (/)  \/  E. y  e.  On  x  =  suc  y ) )  ->  ( A. y  e.  x  ch  ->  ph ) ) )
395, 38syl5bir 210 . . . 4  |-  ( x  e.  On  ->  ( -.  ( Ord  x  /\  -.  ( x  =  (/)  \/ 
E. y  e.  On  x  =  suc  y ) )  ->  ( A. y  e.  x  ch  ->  ph ) ) )
404, 39syl5bi 209 . . 3  |-  ( x  e.  On  ->  ( -.  Lim  x  ->  ( A. y  e.  x  ch  ->  ph ) ) )
41 tfinds.7 . . 3  |-  ( Lim  x  ->  ( A. y  e.  x  ch  ->  ph ) )
4240, 41pm2.61d2 154 . 2  |-  ( x  e.  On  ->  ( A. y  e.  x  ch  ->  ph ) )
431, 2, 42tfis3 4804 1  |-  ( A  e.  On  ->  ta )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649   [wsb 1655    e. wcel 1721   A.wral 2674   E.wrex 2675   (/)c0 3596   Ord word 4548   Oncon0 4549   Lim wlim 4550   suc csuc 4551
This theorem is referenced by:  tfindsg  4807  tfindes  4809  tfinds3  4811  oa0r  6749  om0r  6750  om1r  6753  oe1m  6755  oeoalem  6806  r1sdom  7664  r1tr  7666  alephon  7914  alephcard  7915  alephordi  7919  rdgprc  25373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-tr 4271  df-eprel 4462  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555
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